>> My final thoughts on this is that visualizations are an exciting part of the product, but they are not part of the ingredients.> > See "The World of Blind Mathematcians":> http://www.ams.org/notices/200210/comm-morin.pdf> > Even so, a lot of the famous blind mathematicians seem to think visually. From the article:> "Far from detracting from his extraordinary visualization ability, Morin's blindness may have enhanced it."> > Cheers,> Joe N

That is a better way of putting what I am saying. You cannot "think" visually nor can you embellish a vision with a mathematical sense you do not possess. If I show you two parallel lines that do not intersect and say "See?" that is not the same thing as the mathematical sense that parallel lines never intersect and lines that never intersect are parallel. It will not be there when you mathematically need it. It will not change the way you think. And that is supposed to be the point of all this. To change the way you think. To think mathematically. That is what makes teaching geometry so difficult. The pictures give up the physical truth so quickly that it is nearly impossible for the average student to see the mathematical truth that lies just behind it. It is like asking you to see a star in the middle of the day.

Coincidentally, I have been teaching my son, age 10, algebra this summer. Not only have I been anxiously waiting for this moment to come, but I also have had quite enough of Dan's and Richard's basking in their ability to avoid teaching algebra to their students. Squealing with delight and entertaining themselves with every new way of not teaching they come across. In any event, a non trivial part of my plan involves multiplication, the original topic in this thread.

I'll probably post videos later towards the end of summer, well, unless of course it is a total failure. But the short version of my approach has been to start with cooking recipes with one of the goals being to introduce him to variables and expressions. I had of course thought of starting with the few "formulas" he knows like area and perimeter but it didn't feel right to start with the very notions (formulas) that I was trying to develop to (formulas). And a true feeling for area has a much longer gestation than I originally thought it did. So instead I started with recipes. Recipes have quantities of different things and even have an order in which they must be combined. But the main thing is that they have quantities of different things combined with addition. So if 3 cups of flour, 2 1/2 cups of sugar and 1/4 teaspoon of vanilla make a dozen cookies, how much does it take to make 3 dozen. How many cookies can you make with 1/8 teaspoon of vanilla? How much flour will y!ou need? And so on, of course, abbreviating things till we have 3f or 1/2v and so on. A lot of arithmetic, fractions, variables and expressions. You can understand that if we had two recipes, say one for cookies and one for Kool-aid, and they both required sugar, that we would have to develop some way of figuring out how many cookies to make yet have enough sugar left to make sufficient Kool-aid to wash them down.

This is just the first phase, and there are a few till the end of summer, some leaning more towards the "relation" rather than the "solution". But my goal with regards to multiplication is modest. I want him to really (and I mean really) explain one expression...